On the Number of Arrangements of Pseudolines

Abstract. Given a simple arrangement of n pseudolines in the Euclidean plane, associate with line i the list σi of the lines crossing i in the order of the crossings on line i. $\sigma_i=(\sigma^i_1,\sigma^i_2,\ldots,\sigma^i_{n-1})$ is a permutation of $\{1,\ldots,n\} - \{i\}$ . The vector (σ1 ,σ2, ...,σ_n) is an encoding for the arrangement. Define $\tau^i_j = 1$ if $\sigma^i_j > i$ and $\tau^i_j = 0$ , otherwise. Let $\tau_i=(\tau^i_1,\tau^i_2,\ldots,\tau^i_{n-1})$ , we show that the vector (τ1, τ2, ... , τ_n) is already an encoding. We use this encoding to improve the upper bound on the number of arrangements of n pseudolines to $2^{0.6974\cdot n^2}$ . Moreover, we have enumerated arrangements with 10 pseudolines. As a byproduct we determine their exact number and we can show that the maximal number of halving lines of 10 point in the plane is 13.